Unipotent Packtets for SO(3,2) Map between cells and Special Orbits for big block of SO(3,2) G=SO(3,2) G^v=Sp(6,R) Special Special Orbit Cells Dual Orbit Cells #O_R lambda 5 0 (LDS) 1111 5 1 (0,0) 311 1,2,3 22 2,3,4 3 (1/2,1/2) 11111 4,5 (C,sgn) 4 0,1 2 (3/2,1/2)=rho (221 not special 211 not special) #O_R= number of real forms of dual orbit cell A_q for Sp(4,R) 0: 0 1: 1 2: 2,4 3: 3,5 4: 6 5: 10 ------------------------------------------------------------------- Special Special Orbit Cells Dual Orbit Cells #O_R lambda 5 0 (LDS) 1111 5 1 (0,0) Unique large discrete series of SO(3,2) is stable at (0,0) %stable -d -c 5 -S 1,2 lambda is singular at simple roots: 1,2 cells:5 Parameters (living at lambda): 0 0(0,10): 0 0 [i1,i2] 1 0 ( 2, *) ( 3, 4) Dual parameters (to those living at lambda): 10 10(10,0): 3 3 [r2,r1] 11 10 ( 7, *) ( 8, 9) 1,2,1,2 Dimension of space of stable characters: 1 Everything is stable ------------------------------------------------------------------- Special Special Orbit Cells Dual Orbit Cells #O_R lambda 311 1,2,3 22 2,3,4 3 (1/2,1/2) cells dual cells stable sums 3 2 3+10 2 3 4+11 1 4 1+7 one extra stable sum, for example: 2,3 2,3 3+11 or -3+4 or 1,2 3,4 1+11 or -1+4 etc. AV(cell 2) = real form #1 of 22 AV(cell 3) = real form #2 of 22 AV(cell 4) = real form #3 of 22 sophus-t43:so32-d-new% stable -d -c 2,3,4 -S 1 %stable -d -c 2,3,4 -S 1 lambda is singular at simple roots: 1 cells:2,3,4 Parameters (living at lambda): 1,3,4,7,10,11 1(1,10): 0 0 [i1,ic] 0 1 ( 2, *) ( *, *) 3(3, 8): 1 2 [C+,r2] 5 4 ( *, *) ( 0, *) 2 4(3, 9): 1 2 [C+,r2] 6 3 ( *, *) ( 0, *) 2 7(5, 6): 2 1 [i2,C-] 7 2 ( 8, 9) ( *, *) 2,1,2 10(6, 2): 3 3 [rn,r2] 10 8 ( *, *) ( 5, *) 2,1,2,1 11(6, 3): 3 3 [rn,r2] 11 9 ( *, *) ( 6, *) 2,1,2,1 Dual parameters (to those living at lambda): 11,8,9,6,2,3 11(10,1): 3 3 [r2,rn] 10 11 ( 7, *) ( *, *) 1,2,1,2 8( 8,3): 2 2 [C-,i1] 4 9 ( *, *) (10, *) 1,2,1 9( 9,3): 2 2 [C-,i1] 5 8 ( *, *) (10, *) 1,2,1 6( 6,5): 1 1 [r1,C+] 6 7 ( 0, 1) ( *, *) 1 2( 2,6): 0 0 [ic,i1] 2 0 ( *, *) ( 4, *) 3( 3,6): 0 0 [ic,i1] 3 1 ( *, *) ( 5, *) Dimension of space of stable characters: 5 Basis of stable characters expressed as sums of irreducibles 1,3,4,7,10,11: 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 -1 0 1 0 0 0 -1 1 0 0 0 0 ------------------------------------------------------------------- Special Special Orbit Cells Dual Orbit Cells #O_R lambda 11111 4,5 (C,sgn) 4 0,1 2 (3/2,1/2)=rho trivial, sgn are stable at rho sophus-t43:so32-d-new% stable -d -c 0,1 %stable -d -c 0,1 cells:0,1 Parameters (living at lambda): 8,9 8(6, 0): 3 3 [r2,r2] 9 10 ( 7, *) ( 5, *) 2,1,2,1 9(6, 1): 3 3 [r2,r2] 8 11 ( 7, *) ( 6, *) 2,1,2,1 Dual parameters (to those living at lambda): 0,1 0( 0,6): 0 0 [i1,i1] 1 2 ( 6, *) ( 4, *) 1( 1,6): 0 0 [i1,i1] 0 3 ( 6, *) ( 5, *) Dimension of space of stable characters: 2 Basis of stable characters expressed as sums of irreducibles 8,9: 0 1 1 0